We propose to develop a mathematical theory to calculate the conformational entropy for a chain molecule with given contact constraints. We will use the new entropy calculation method to compute protein stabilities and compare against experiments. This project involves 5 steps: (1) Using graph theory, we calculate the conformational entropy for chain molecules with any type of contact maps. (2) We take the set of small proteins that have been studied by calorimetry. (3) We choose some energy function. It is simplest to start with hydrophobic and polar terms, or augment an HP model with hydrogen bonding interactions, for which there are now good estimates of the energies from model studies. Or we can use the estimates from earlier mean-field models, which are already modestly successful for myoglobin. If we reach a stage where the energy function is limiting (which we can determine from the calorimetric enthalpies and heat capacities), we can change the energy function in later iterations. (4) For the native structure, known from the PDB, we will use the given energy function to compute the native free energy. (5) For each amino acid sequence, we will generate an ensemble of denatured conformations, by hydrophobic zippers for example, then multiply by the Boltzmann factors (determined by the number of HH contacts, and H-bonds, as appropriate), compute the density of states from the matrix method, and sum to get the denatured state partition function. This method will give enthalpies, entropies, heat capacities, and free energies to compare with experiments.